Question

A question in Probability & Statics:-

At a checkout counter, customers arrive at an average of 1.5

per minute. Assuming poisson distribution

1- The probability of no arrival in two minutes is.....

2- The variance of the number of arrivals in two minutes

is.....

3- Suppose X has binomial distribution with n=1000 and p

=0.002 , Then P(x=3)=....

Answer 1

1. $X\sim Po(\lambda t), \lambda t=1.5(2)=3$

$$P(X=0)=\dfrac{e^{-3}(3)^0}{0!}=0.049787$$

2.

$$Var(X)=\lambda t=3$$

3. When the value of $n$  in a binomial distribution is large and the value of $p$ is very small, the binomial distribution can be approximated by a Poisson distribution. If $n > 20$ and $np < 5$ or $nq < 5$ then the Poisson is a good approximation.

Check

$$n=1000>20, np=1000(0.002)=2<10$$

Then $\lambda=np=2$

$X\sim Po(2)$

$$P(X=3)=\dfrac{e^{-2}(2)^3}{3!}$$ $$=0.180447$$